Three-Node Shared-Edge Network under Double-Ended Correlated Load

We study a three-node network with six independent directed edges and double-ended correlated load. Under node-permutation symmetry S_3, the twelve-dimensional state space decomposes as R¹²=3·1⊕ 4· Vstd. The trivial sector reduces to one strictly negative scalar and one two-dimensional coupled block. In the non-orthogonal uniform-edge coordinate δ w the trivial block is M_0ᵈʳᵃᶠᵗ; in the orthogonal coordinate compatible with the modal transformation T it is M_0ᵒʳᵗʰ, related by the similarity M_0ᵒʳᵗʰ=D_0M_0ᵈʳᵃᶠᵗD_0⁻¹, D_0=diag(1,1,√2), with common characteristic polynomial z^2+(r-s)z-rs-4λ u. Hurwitz conditions _1-_2<0, r>s, rs+4λ u<0, where s=δ/2+4w^*(χ-ψ), r=2ρ w^*, u=w^*(α+β). The sign sector supports an exact one-dimensional equilibrium family; the zero eigenvalue is forced by that family, and the S_3 decomposition identifies the tangent vector as lying in the sign sector. Under isotropic diffusion η_A=η_B=ηdiff, the standard sector in the non-orthogonal (x,y) coordinate gives the exact orthogonal reduction QMstd⁽⁴⁾Q^T=tₐₙₜᵢ⊕ M_3, with tₐₙₜᵢ=_1-_2-3ηdiffw^* and M_3={pmatrix} ssym&-λ/√2&-λ/√2\\ -√2w^*(α+ζ)&-2ρ w^*&ρ w^*\\ -√2w^*(β-ζ)&0&0 {pmatrix}, ssym=δ/2-3ηdiffw^*-2w^*(χ+2ψ). The characteristic polynomial is P_3(z)=z^3+(r-ssym)z^2-(rssym+λ u)z-32λ rw^*(β-ζ), obtained by direct cofactor expansion of (zI_3-M_3); the coefficient -32λ rw^*(β-ζ) matches the value 3/40 at the admissible parameter point. At the admissible parameter point with γ=2/15>0, numerical evaluation of TJT^T confirms that isotypic cross-blocks vanish, that the trivial and sign blocks take their analytic forms, and that the full characteristic polynomial is z(z+13/6)(z^2+397/10z+39/10)(z+37/15)^2(z^3+40z^2+174/25z+3/40)^2, with algebraic multiplicity of zero exactly one. For the four copies of the standard representation, the intertwiners U_A=U_B=I_2, U_s, and U_a are constructed by solving the intertwining equations for the generators (12) and (123). Under the standard cycle identification, ρ_a(r)=ρref(r) and the two valid U_a matrices differ by an overall sign, U_a'=-U_a. The aligned form Jaligned=I_2⊗ Mₘᵤₗₜ is verified numerically at the parameter point; the elementwise similarity Mstd⁽⁴⁾=SMₘᵤₗₜS⁻¹ is stated as the remaining open computation. A conditional nonlinear statement (Proposition 7.1) is proved under (H1)–(H4): F∈ C^2, local smoothness of the equilibrium family, algebraic multiplicity one at zero, and strict transverse spectral gap at the base point. Then locally the family is a center manifold; the reduced dynamics satisfies a=0 identically; the family is locally transversely attracting; no single equilibrium is asymptotically stable as an isolated point. The phase shift a_∞-a(0) is not quantified. All results are linear spectral analysis, local nonlinear center-manifold analysis, or one-point numerical evaluation.